1. Field of the Invention
The present invention relates to mathematical modeling techniques and, more particularly, the process of visually fitting one or more curves to a limited set of empirical data for the interpolation or extrapolation of a complete predictive data set and, even more particularly, to a simplified means for describing and predicting the thermal behavior of an item subjected to conditions meant to induce heat transfer within the item.
2. Description of the Background
The basic concept of a thermal model works through mathematical manipulation of equations with the purpose of matching a predetermined thermal response. Conventional thermal models are based upon the fact that an object at an initial temperature (T1), exposed for predetermined period of time to an environmental temperature (Tenv.), the temperature of the object at the end the time period (T2) will be some temperature between T env. and T1. The basic equation is as follows:T2=T1+(Tenv.−T1)×F Where:
F=constant fraction (a number between 0 and 1)
This basic principle works well to represent the skin (i.e. outer layer) response of any object or to represent objects that are small and have a high thermal conductivity (like a metal wire). However, when the object in question has a large thermal mass and a complicated internal configuration with different materials, the thermal response of the inside parts lags the skin response and the mathematical equations describing the thermal response are not easy to develop.
Previously, an engineer needed to know a great deal about an object in order to generate a mathematical model describing, or predicting, its thermal behavior. The information set had to include the size, shape, and weight of the item, as well as certain material-of-construction properties such as, but not limited to, heat capacity, thermal conductivity, and heat transfer coefficient. Only then could a mathematical model of the item's thermal response be generated. The model then required validation with an instrument-laden version of the item in question. This had to be done to demonstrate the correlation between the values of temperature vs. time predicted by the model as compared to the actual values generated during testing. While the rise of computer technology has aided the process significantly, the creation of the initial model has always been a very cumbersome process requiring a great deal of time.
The present inventor is not the first to address the prediction of thermal behavior in an object. For example, U.S. Pat. No. 6,164,816 to Aderhold et al. discloses a technique and system for tuning temperature sensor readings in a thermal processing chamber that includes determining an actual temperature profile for a substrate based on measurements of the substrate. A simulated temperature profile for the substrate is calculated using a respective interim temperature correction value for one or more temperature sensors associated with the chamber. A Gaussian-like distribution for thermal contributions from multiple radiation sources in the chamber is used to simulate the temperature profile. The simulated temperature profile and the actual temperature profile are combined to form an estimated temperature profile. A final value for each respective temperature correction value is determined using an optimization algorithm which results in the estimated temperature profile being substantially uniform across the surface of the substrate. Each final temperature correction value is used as an offset to temperature measurements obtained from the corresponding temperature sensors. Unfortunately, the Aderhold et al. technique/system only generates a series of temperature correction values, it does not provide a thermal behavior model (i.e. each temperature correction value is used as an offset to actual temperature measurements obtained from a series of temperature sensors).
Additionally, generic curve fitting software is well-known. For example, a software application entitled “Curve Fitting Toolbox” is commercially-available from The MathWorks, Inc. of Natick, Mass. (see “www.mathworks.com/products/curvefitting”). Unfortunately, significant re-programming effort and knowledge would be required to modify/customize a generic software application, such as that identified above, for use in thermal behavior modeling. Even then, the modified/customized application would not readily provide the user-friendly visual controls required to manually approximate non-linear regression by manipulating a set of generic curves to fit a set of actual measured curves.
Therefore, there remains a need for a thermal behavior modeling technique designed to predict the thermal response of an item, subjected to conditions meant to induce heat transfer within the item, via interpolation or extrapolation from a limited set of actual temperature data. The modeling technique should not require the copious amounts of data (e.g. item size, shape, and weight, material-of-construction heat capacity, thermal conductivity, and heat transfer coefficient) needed heretofore. The modeling technique should be simple to use through manipulation of a series of visual controls and sufficiently flexible (mathematically) for use in a variety of thermal response applications. Finally, the modeling technique should not require the use of proprietary software, but be capable of implementation using any readily available spreadsheet software application (e.g. Microsoft® Excel®) to provide for widespread use.